Electric Field Strength- variations

If the electric field lines are curved as opposed to straight, say from straight parallel lines parabolic curves develop, would there be any change in the field strength? ie would it be stronger when the lines are parallel than when they curve?

i was trying to describe a diagram that I didnt quite understand and wasn't sure if the electric field strength would be smaller when curved.

It looks a bit like the path an electron would take in a uniform electric field. With the direction of the electron being initially perpendicular to the direction of the uniform field between the plates. The electron would then follow a parabolic curve. But on this diagram only the electric field lines are present, no plates etc. more or less showing the direction of electric field lines. Like the tangent at the curve, showing direction.

What i wanted to know, was if E is stronger when the lines are parallel as opposed to when then move off into the curve.

Staff: Mentor

The strength of the E-field is indicated by how closely spaced the field lines are, not their curvature.

E-field lines are always normal (perpendicular) to conductors at the surface, because a conductor cannot support tangential electric surface fields. So field lines at curved surfaces are always curved.
Bob S

[added] If the field lines originate from a concave surface of a conductor, they are leaning toward each other. In this case, is the electric field a maximum at the surface, or away from the surface, such as near the center of curvature for the surface? What about requirements due to the orthogonality of the electric field lines and the equipotential lines, analyticity, Cauchy theorem, Cauchy Riemann equations?
Bob S

Staff: Mentor

ok thanks for your help...still not sure what this diagram means though.

where the field lines move off into the curve, an annotation says E (which i assume is the electric field strength) is small and where the field lines follow the initial straight path, where they lie parallel to each other, the annotation says E is large.

This reference says the width between the field lines is indicative of field strength, but note, this is relative to a diagram of a particular problem and the field strength or density is really determined by a calculation and can't be derived from a diagram.

This reference says the width between the field lines is indicative of field strength, but note, this is relative to a diagram of a particular problem and the field strength or density is really determined by a calculation and can't be derived from a diagram.

I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?
Bob S

Staff: Mentor

ok thanks for your help...still not sure what this diagram means though.

where the field lines move off into the curve, an annotation says E (which i assume is the electric field strength) is small and where the field lines follow the initial straight path, where they lie parallel to each other, the annotation says E is large.

What is the spacing between the lines where the annotation says E is large? Are they close or far compared to where the annotation says it is small?

[added] If the field lines originate from a concave surface of a conductor, they are leaning toward each other. In this case, is the electric field a maximum at the surface, or away from the surface, such as near the center of curvature for the surface?
Bob S

In case of a concave surface the field lines are going to be wider spaced, a lot will end up on the sharper edges of the plate and towards the convex side. To see that this is so increase the curvature of the concave side until your surface forms a hollow cylinder or hollow sphere. There will be no field lines inside those at all.

I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?
Bob S

I don’t think so. Field lines tend to spread out sideways so as to minimise the stored energy per unit volume.

From Bob S
I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?

I agree that the space between field lines is indicative of field strength. For a specific situation, can the electric field be stronger in a space (without free charges) than at a conducting surface where the field lines terminate on surface charges?
Bob S

Field and Wave Electromagenetics by David K. Cheng (1985).

Boundary Conditions at Conductor / Free Space Interface

[tex]E = \frac{\rho_{s}}{\epsilon_{0}}[/tex]

"The normal component of the E field at a conductor-free space boundary is equal to the surface charge density (rho) on the conductor divided by the permitivity of free space."

Finding the E field at a point P located at radius R from a differential surface element ds is calculated by taking a surface integral (double integral notation not shown):

where this should converge to the boundary condition specified above at R = 0, but I'm not up to date on my double integral techniques and it appears to me that E might blow up to infinity as R approaches zero in the integral evaluation?

In any case E should decrease with an increase of R away from the conductor surface.

Also see Gauss's law in reference to this thread, where the flux of the normal E field is constant for any surface enclosing a charge, thus as the radius to the surface increases, the normal flux density decreases. For a symmetrical problem the E field lines are easily visualized as being farther apart as the radius increases of the enclosing surface.

where this should converge to the boundary condition specified above at R = 0, but I'm not up to date on my double integral techniques and it appears to me that E might blow up to infinity as R approaches zero in the integral evaluation?

In any case E should decrease with an increase of R away from the conductor surface.

Also see Gauss's law in reference to this thread, where the flux of the normal E field is constant for any surface enclosing a charge, thus as the radius to the surface increases, the normal flux density decreases. For a symmetrical problem the E field lines are easily visualized as being farther apart as the radius increases of the enclosing surface.

Does this apply to the fields in a parallel-plate capacitor, where the field lines are parallel and equally dense everywhere, both at the plates, and everywhere in between?
Bob S

If the permitivity (dielectric constant) is that of free space, I believe the boundary conditions and integral above would give the constant field of a parallel plate capacitor after applying superposition of the two E fields upon the space between the plates. Of course in a practical capacitor the uniform field assumes plate spacing is small and area large, so the field is constant near the center. Changing the dielctric permitivity requires similar relations defined for the displacement vector D, according to Cheng. I don't know if this helps with your question?